a. Overview

A 2D model of the velocity field of the bottom boundary layer on the continental shelf, in the presence of wave motions, is developed to demonstrate wave bias and to provide a framework for the presentation of techniques for removing it. Although a simplification, a 2D analysis—in which the direction of the turbulent shear stress, the wave motions, and the components of the instrument coordinate system lie in a vertical plane—captures the important features of the wave bias problem. The 2D analysis presented here is a simplification of Trowbridge (1998), although it has been expanded to include the effects of waves on heat flux estimates and it relaxes assumptions about near-equality of wave-induced motions at the spatial scales of interest.

The analysis relies on several assumptions. The most fundamental is that the ratio of the spatial coherence scale of the wave-induced fluctuations to the spatial coherence scale of the turbulence-induced fluctuations is large, so a spatial separation exists at which the wave-induced fluctuations are coherent and the turbulence-induced fluctuations are incoherent. We also require that the wave- and turbulence-induced fluctuations are incoherent with one another (a justifiable assumption, because waves and turbulence are essentially independent processes outside of a thin wave boundary layer), and that the statistical properties of the waves and turbulence are stationary. For the purpose of illustration, we assume that the velocity sensors have perfect response, all components of the turbulent covariance tensor have equal magnitudes [which is a good assumption for boundary layer flows (e.g., Tennekes and Lumley 1972)], the waves are of small amplitude and narrow-banded in frequency, and stratification is due to a uniform vertical temperature gradient.

The horizontal component x of the model position vector x = (x, z) lies in the common direction of the wave propagation and the shear stress, and the vertical component z is defined to be positive upward with z = 0 at the bed (Fig. 1). The model velocity vector u = (u, w) where u and w are the horizontal and vertical components of the velocity, respectively, is composed of contributions from, respectively, mean current, waves, and turbulence: u = u + ũ + u′. The temperature similarly is denoted by T = T + T̃ + T′. The wave-induced component of the fluctuations can include contributions from both surface and internal waves, although we do not consider interactions between them. The quantities of interest are the turbulent shear stress −ρu′w′ and the turbulent vertical heat flux ρc_pT′w′, where ρ is density and c_p is heat capacity.

In the remainder of this section, we first illustrate the problem of wave bias that results from a simple application of the direct eddy correlation technique to measurements obtained with an instrument that is tilted with respect to the principle axes of energetic surface wave motions (section 2b). We demonstrate that taking the difference of records from spatially separated sensors can be used to reduce the wave bias (section 2c). Then, in contrast to the previous estimates that assume the wave motions are essentially equal at the two positions, we introduce a linear filtering technique that further reduces wave bias by accounting for small differences in wave amplitude or phase between the two sensors (section 2d). Finally, we present an empirical constraint on the magnitude of the vertical separations that satisfies the critical assumption that the turbulent fluctuations at the two locations be uncorrelated (section 2e).

b. The wave bias problem

An estimate of u′w′ using the velocity U = (U, W), measured in instrument coordinates that are rotated a small angle θ from the model coordinate system (Fig. 1) to first order in θ, is

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In a similar way, an estimate of T′w′ to first order in θ is

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The first term on the right side of (1) and (2) is the desired vertical turbulent flux of horizontal momentum or heat, respectively. The second term is a “real” wave bias, representing vertical transport by waves. The third and fourth terms are “apparent” turbulence- and wave-induced biases arising from the orientation of the instrument coordinate system. The apparent turbulence bias is the well-known instrument leveling error (Pond 1968).

For the case of gentle bottom slopes, small rotation angles, and irrotational waves, the turbulence bias in (1) is small, but the wave biases can be an order of magnitude larger than the desired turbulent momentum flux (Trowbridge 1998). Observations in the atmospheric boundary layer indicate that the components of the heat flux vector have the same order of magnitude (Kaimal et al. 1972), indicating that the turbulence bias in (2) is small in comparison with the desired covariance if θ is small. We can estimate the real wave bias in (2) as follows. For the case of a uniform vertical temperature gradient and small-amplitude motions, the wave-induced temperature fluctuations arise from advection of the vertical temperature gradient and, for progressive waves, may be estimated as

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where c is the wave phase speed. T′w′ may be estimated roughly as −κzu∗dT/dz (e.g., Businger et al. 1971), where u∗ is the bottom friction velocity and κ (=0.4) is von Kármán's constant. With these estimates, the ratio of wave-induced bias to turbulent heat flux is

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indicating that for near-bottom values typical of the outer shelf for surface waves (ũ² = 0.01 m² s⁻², c = 20 m s⁻¹, u∗ = 0.01 m s⁻¹, and θ = 0.01), the wave-induced bias is 3 orders of magnitude smaller than the turbulent heat flux.

If the measured velocity vector can be rotated into the principal axes of the wave-induced velocity field, the wave biases are removed. In practice, the orientation of the principal coordinates of the wave-induced velocity field usually is not known with sufficient accuracy to remove wave bias by rotation. The objective of the present analysis is thus to consider alternative techniques for removing the real and apparent wave biases from estimates of the turbulent covariances u′w′ and T′w′ in (1) and (2).

c. Differencing strategies

Trowbridge (1998) showed that the wave bias in estimates of momentum flux can be reduced to an acceptable level by taking the difference of (“differencing”) measurements obtained from two sensors separated by a distance large in comparison with the correlation scale of the turbulence but small in comparison with the inverse wavenumber of the waves, assuming that the correlation scale of the waves is much greater than the correlation scale of the turbulence and that the waves and turbulence are uncorrelated. Here, a slightly different approach in which only one of the measurements composing the covariance is differenced, is used. This approach is beneficial because the resulting covariance estimate corresponds to the position of a single sensor, rather than an average of the covariances at each sensor, and two estimates are available for each sensor. The resulting turbulence estimates are not independent, but the waves are removed with essentially independent information, so the redundant estimates provide a check on the technique.

In the following, a parenthesized subscript is used to identify the position at which a particular measurement is obtained. For example, u₍₁₎ is the horizontal velocity measured at the location of the first sensor. The operations of differencing and averaging between two sensors are represented by an upper-case delta sign and angular brackets, respectively. For example, Δu and 〈u〉 are the difference and average, respectively, of the horizontal velocities measured by a pair of sensors. We focus here on the differencing of velocity measurements.

We consider three estimates of u′w′ given measurements at a pair of sensors: (1/2)cov(ΔU, ΔW), cov[ΔU, W₍₁₎], and cov[U₍₁₎, ΔW]. The assumption that wave- and turbulence-induced velocities are uncorrelated (justified in section 2a), immediately allows us to decompose contributions to the u′w′ estimates into turbulence and wave components.

Assuming the turbulent fluctuations are uncorrelated at the two locations (discussed further in Section 2e), the turbulent component of the three estimates of u′w′ can be written (to first order in the small angles θ₁ and θ₂; see Fig. 1) as

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The first term on the right side of (5) is an average of the turbulence-induced covariance at each of the sensors, and the first term on the right side of (6) and (7) is the desired turbulence-induced covariance at the location of the first sensor. The second term on the right side of (5)–(7) is a small apparent turbulence bias.

The wave component of the three u′w′ estimates may be written (also to first order in θ₁ and θ₂) as

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Wave contributions to the measured covariance arise from differences in wave-induced velocity between the two locations and differences in alignment between the two sensors. We note that, in contrast to (5)–(7), (8)–(10) contain correlation terms between the two positions and that the differencing techniques, in principle, would work perfectly if the wave-induced fluctuations at the two locations were equal. There obviously is a compromise in choosing the separation distance between having the turbulence uncorrelated and the waves highly correlated.

To illustrate the magnitudes of the terms in (8)–(10), we restrict the scope of the analysis to the measurement of turbulence in the presence of surface waves with a pair of vertically separated sensors. Differenced quantities are represented with first-order expansions. For example, Δu ≃ r∂ũ/∂z, where r is the separation between the sensors and ∂ũ/∂z is evaluated at a location between the two sensors for the estimate (1/2)cov(ΔŨ, ΔW̃) and at the location of the first sensor for the estimates cov[ΔU, W₍₁₎] and cov[U₍₁₎, ΔW]. Also, we assume that kz ≪ 1, where k is the wavenumber of the waves, allowing the hyperbolic functions describing the vertical variation of wave-induced velocity to be represented with first-order expansions. With these assumptions, (8)–(10) can be approximated as

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For near-bottom measurements, we expect that terms like ũw̃, θũ², and Δθũ² are of comparable magnitude and much larger than terms such as θw̃² and Δθw̃². All of the large terms on the right side of (11) and (12) have been reduced by at least the factor k²rz, while two of the large terms in (13) have only been reduced by the factor r/z, which is constrained by the assumption that the turbulence must be uncorrelated at the two positions to be of order 1 (see section 2e). The stress estimate (1/2)cov(ΔŨ, ΔW̃) has a greater reduction in wave bias at the cost of averaging the turbulent covariances between the two locations. For near-bottom measurements, the reduction in apparent wave bias resulting from the rotation of horizontal velocity into the vertical is better for (1/2)cov(ΔŨ, ΔW̃) by an additional factor of k²rz.

d. Further reduction in wave bias with adaptive filtering

Failure of the differencing technique described in section 2c occurs under conditions of high wave energy if differences in amplitude or phase of the wave-induced motions between the two locations cause the difference terms Δũ or Δw̃ appearing in (8)–(10) to be significant. A solution is to minimize the difference terms with linear filtration techniques. Assuming that the wave-induced fluctuations are completely spatially coherent, least squares filtering can be used to estimate the coherent component of velocity at one position with velocity measurements at the second position. Thus, the assumption behind the simple differencing technique used earlier—that the wave-induced motions at the two sensors are essentially identical—is replaced by the assumption that the wave-induced motions at the two sensors are essentially perfectly coherent. This relaxation allows potential differences in wave phase and amplitude that contribute to the difference terms in (8)–(10) to be minimized. In other words, because the coherence scale of wave motions is expected to be greater than the correlation scale, the filtering technique extends the range of application of the differencing techniques. The assumption that the turbulence is spatially uncorrelated is replaced by the assumption that the turbulence is spatially incoherent.

Here, the analysis is no longer limited to vertical separations. We begin by using the assumption that the wave motions at the two positions are coherent to establish a linear functional relationship between the wave-induced fluctuations measured at positions (1) and (2). For example, the wave-induced horizontal velocities are assumed to be related by

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Here, t is time and h(t) is a filter that represents the relationship between the wave-induced fluctuations at the two locations. In effect, (14) states that if Ũ₍₁₎ and Ũ₍₂₎ are perfectly coherent, then Ũ₍₁₎ is completely predictable from Ũ₍₂₎. The goal of this analysis is to use the total measured velocities to estimate h(t). A more nearly wave-free estimate of horizontal velocity at position (1) than ΔU = U₍₁₎ − U₍₂₎ is then ΔÛ = U₍₁₎ − Û₍₁₎, where

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is the estimated wave-induced velocity at position (1), using estimates of the filter weights ĥ(t) and the measured velocities at position (2). The estimated wave velocity Û₍₁₎ contains a turbulence component, but it is of no consequence if the assumption that the turbulence is spatially incoherent is valid. If the estimates cov[ΔU, W₍₁₎] and cov[U₍₁₎, ΔW] are replaced by cov[ΔÛ, W₍₁₎] and cov[U₍₁₎, ΔŴ], respectively, the problematic difference terms appearing in (9) and (10) are minimized to the extent that the model assumptions are valid.

The filter weights in (14) are estimated by finding the ordinary least squares solution of a transversal filter model (e.g., Haykin 1996) that has been modified to a noncausal form for postprocessing purposes:

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Here, A is an M × N windowed data matrix of velocity at position (2), where M is the number of data points and N is the number of filter weights (N must be odd for the filter to be symmetric), h is a vector of filter weights, and U₍₁₎ is a vector of position (1) velocity. The mth row of A is [u(m − (N − 1)/2), … , u(m), … , u(m + (N − 1)/2)], where u(m) is the mth discrete sample of u. The solution is

ĥA^TA⁻¹A^TU₍₁₎(17)

and estimates Û₍₁₎ of the wave-induced velocity at position (1) are found by convolving the measured velocity record with the estimated filter weights

Û₍₁₎Aĥ(18)

In practice, we have encountered some complications in applying the filtering technique described above (to be discussed below); in general, however, robust estimates of the filter weights are obtainable with the adjustment of only two significant parameters: the number of data points M and the number of filter weights N.

e. Turbulence correlation lengths

For these wave bias removal techniques to work in practice, the sensor separation must be large relative to the correlation length scale of the turbulence so that the turbulence cross-correlation terms u^′₍₁₎u^′₍₂₎, u^′₍₁₎w^′₍₂₎, w^′₍₁₎u^′₍₂₎ neglected in (5)–(7) (by the assumption that the turbulent fluctuations at the two positions are uncorrelated) are small. It is easily shown that all of the cross correlation terms except u^′₍₁₎w^′₍₂₎ are multiplied by θ, and therefore, assuming that the elements of the turbulent correlation function tensor have the same order of magnitude, we only need to ensure that u^′₍₁₎w^′₍₂₎ is small as compared with u^′₍₁₎w^′₍₁₎. Because of the strong intermittency of turbulent eddies, we expect that the turbulent coherency scale is nearly equal to the turbulent correlation scale.

We have established an empirical guideline ensuring that u^′₍₁₎w^′₍₂₎ is small by considering the off-diagonal component of the turbulent correlation function tensor obtained in an estuarine bottom boundary layer. Trowbridge et al. (1999) reported the deployment in August of 1995 of a quadrapod on the bed of the Hudson River adjacent to Manhattan, New York, for two weeks with a vertical array of five Benthic Acoustic Stress Sensor (BASS) current meters (see section 3) at heights of 0.3, 0.6, 1.2, 2.1, and 2.4 m above the bottom in a wave-free turbulent boundary layer. The instruments were sampled in bursts of 3.28 min at 6.25 Hz. To evaluate the importance of u^′₍₁₎w^′₍₂₎, we calculated representative cross-correlation functions by taking the average over a period of five days of R_uw = u(z)w(z + r) calculated from the 3.28-min bursts of data (Fig. 2). The cross-correlation functions indicate that the ratio of the vertical separation to the height of the lower sensor r/z must be greater than approximately 5 for the turbulence to be considered uncorrelated, that is, less than one-tenth of the value at zero separation. In this particular environment, the cross-correlation function decreases more rapidly as z increases, possibly because the scale of the eddies is limited by factors other than height above bottom (Trowbridge et al. 1999).

a. Conditions at the CMO site

Before proceeding to a test of the proposed techniques for removing wave bias, it is worthwhile to briefly describe the conditions at the CMO field site during the first deployment. Here, we describe the temporal variability of burst statistics and we present spectra representative of conditions of energetic surface waves and conditions of energetic internal motions.

Time series of near-bottom mean currents at the CMO site (Fig. 5a) are usually dominated by a rotary semidiurnal tide with an amplitude of approximately 0.1 m s⁻¹. Subtidal events consist of predominantly westward flows with velocities up to 0.2 m s⁻¹. Time series of mean sound speed measured by the top and bottom BASS sensors (Fig. 5b) indicate that the bottom boundary layer at the CMO site usually is well mixed in sound speed. Occasionally, however, near-bottom stratification is strong, resulting in sound-speed differences between 0.38 and 5.50 m above the bed of up to 10 m s⁻¹, which roughly corresponds to 5°C. Time series of the standard deviation of horizontal velocity σ²_u + σ²_υ (Fig. 5c) contain contributions from surface waves, internal motions, and, to a much lesser extent, turbulence. Surface waves generated by the close passage of Hurricane Edouard, a nor'easter, and Hurricane Hortense are visible as peaks in σ²_u + σ²_υ centered on days 246, 251, and 258, respectively. The intermittent peaks of short duration are associated with internal motions (often simply referred to as internal waves hereinafter) with periods on the order of 5–15 min (including trains of wavelike motions as well as more borelike features), as determined from the inspection of individual burst time series (an example of a borelike feature is presented below).

Representative spectra of longitudinal and vertical velocity fluctuations, S_uu and S_ww, during conditions of strong surface waves (Fig. 6a) contain an energetic surface-wave peak centered at 0.07 Hz, corresponding to a wavelength greater than 300 m, and a phase speed of 20 m s⁻¹. The spectral levels of the S_uu wave peak are more than 2 orders of magnitude larger than the underlying stress-carrying eddies. The corresponding spectrum of temperature fluctuations, S_TT (Fig. 6b), contains a small peak at the dominant surface-wave frequency, consistent with the estimate (3), that is, the idea that T̃²/T′² is small when compared with ũ²/u′².

Representative spectra of longitudinal and vertical velocity fluctuations during conditions of strong internal waves (Fig. 7a) contain a low-frequency “hump” in S_uu below 0.002 Hz. Between 0.002 and 0.01Hz, the spectral levels of the horizontal velocity fluctuations fall off rapidly, giving the spectrum an overall concave upward shape. S_TT also contains energetic low-frequency variability (Fig. 7b), but, in contrast to S_uu, the temperature spectral levels do not fall off as rapidly between 0.002 and 0.01 Hz.

b. Shear stress

It is instructive to consider the cospectra that are integrated to yield the three sets of shear stress estimates described in section 3, because wave-induced contributions can be recognized as deviations from the form expected in typical boundary layer flows. For reference, we present empirical forms for Co_uw and the running integral of Co_uw, ∫^f₀ Co_uw(f′) df′ (known as an ogive curve) that were determined from measurements in the wall region of the atmospheric boundary layer (Kaimal et al. 1972) [and were shown to be consistent with cospectra obtained in the wall region of shallow-water, tidal-bottom boundary layers (Soulsby 1977)] in Fig. 8a. The energetic eddies responsible for transmitting stress lie approximately in the range of apparent nondimensional wavenumber 0.1 < 2πfz/V < 10 with a peak near 2πfz/V = 1, in which V is the burst-averaged, along-stream velocity and 2πf/V is the apparent wavenumber of the turbulent eddies advected past a point sensor by V. When energetic waves are present, this form of Taylor's hypothesis is inadequate because advection of eddies by the wave motions is important (Lumley and Terray 1983) and the actual wavenumber distribution is different than that given by 2πf/V. We present the results in the form of ogive curves, which are essentially low-pass-filtered cospectra, because cospectral estimates are inherently noisy.

To illustrate the filtering process, we present an example of the estimated filter weights between elevations of 5.4 and 1.1 m above the bed [positions (2) and (1), respectively] during energetic surface-wave conditions (Fig. 9) that demonstrates that the filters are physically meaningful. For clarity, we use a notation h_ij, where i denotes the component of velocity that is the output of the filter and j denotes the component of velocity that is the input to the filter. The estimation of Ũ₍₁₎ is dominated by h_uu (Fig. 9a), which has a maximum at zero lag, as expected for vertically separated instruments, and decreases with increasing lag, with a zero crossing near the fourth lag, or about 3.3 s, which is roughly consistent with a dominant surface-wave period of 15 s. The weights of h_uυ and h_uw are near zero, indicating that V₍₂₎ and W₍₂₎ are not important inputs in the estimation of Ũ₍₁₎, although it is reassuring that they take near-zero values. In contrast, the estimation of W̃₍₁₎ is not dominated completely by h_ww (Fig. 9b), and the weights of h_uυ and h_uw are significant, with skew-symmetric peaks near the third lag, indicating that U₍₂₎ and V₍₂₎ are coherent with W̃₍₁₎ but are approximately in quadrature, as expected.

As an example of the effect of moderately energetic surface waves on Co_uw, we present a spectrum of horizontal velocity and cospectral ogive curves at 0.70 m above the bed, position (1), for a burst in which the surface-wave rms velocity equals 0.07 m s⁻¹ (Fig. 10). The surface-wave spectral peak (Fig. 10a) is nearly collocated with the expected maximum in turbulence cospectral density (see Fig. 8a). The raw covariance estimate has a large contribution (visible as the region of steep slope in the ogive curve) at the frequency of the surface waves (Fig. 10b). This is anomalous for boundary layer turbulence (compare with Fig. 8a) and is attributable to wave contamination. Of the two differenced estimates, cov[ΔU, W₍₁₎] contains little or no surface-wave contamination, while cov[U₍₁₎, ΔW] contains a surface-wave contribution of magnitude roughly equal to the raw estimate (Fig. 10c), consistent with expectations of section 2c. Neither of the filtered estimates, cov[ΔÛ, W₍₁₎] or cov[U₍₁₎, ΔŴ], contains a net contribution to the shear stress from surface waves (Fig. 10d). There is some contamination of Co_UΔŴ. However, the filtering has compensated in a manner that yields no net contribution to cov[ΔÛ, W₍₁₎].

As a further test of the techniques in removing surface wave contamination, we present an example of the effect on Co_uw of highly energetic surface waves (rms velocity equal to 0.12 m s⁻¹ at 0.70 m above the bottom; (Fig. 11). As in the example for moderate surface-wave energy, the estimates cov[U₍₁₎, W₍₁₎] and cov[U₍₁₎, ΔW] fail to remove wave contamination (Figs. 11b,c), whereas the estimates cov[ΔÛ, W₍₁₎] and cov[U₍₁₎, ΔŴ] succeed at removing surface-wave contamination (Fig. 11d). In contrast, the differenced estimate cov[ΔU, W₍₁₎] also fails to remove surface-wave contamination (Fig. 11c), indicating that the filtering technique is necessary to remove wave contamination when highly energetic surface waves are present.

An example burst time series of longitudinal and vertical velocity 1.10 m above the bottom on day 253, during a period of measurable stratification (see Fig. 5b) and most likely associated with a horizontal intrusion, captures the passage of an energetic internal motion with near-bottom velocities greater than 0.1 m s⁻¹ and a duration of approximately 5 min (Fig. 12a). The sound-speed record for this burst also records the passage of the “event” (see Fig. 15, described in next section) and indicates that the event had a borelike character. The internal motion made a large low-frequency contribution to the raw cospectrum below 2πfz/V = 0.02, and although the filtered estimate is better than the raw estimate, in contrast to the case of surface waves, the proposed techniques are not successful at removing the observed wave bias (Fig. 12b). The failure is partially owing to the long duration of the internal motion as compared with the period of surface waves and the length of the burst and to the intermittent nature of the internal motion. There is also a small surface-wave bias visible in the ogive curve of Co_ΔÛW (similar to those identified in Figs. 10 and 11), suggesting that the use of constant filter weights during a burst with nonstationary internal wave properties degrades the ability of the filtering technique to remove surface-wave bias. In practice, the range of internal motion contamination is nearly out of the expected nondimensional wavenumber range of the energy-containing eddies (Fig. 8a) and the unwanted internal motion bias can be removed by windowing Co_uw.

A comparison of u′w′ estimates from the entire 6-week deployment using cov(ΔÛ, W) as a standard demonstrates that the raw shear stress estimates cov(U, W) are contaminated with surface-wave-induced covariance during the energetic surface-wave events (Fig. 13a). As anticipated in section 2c, the differenced shear stress estimate cov(ΔU, W) is mostly free of surface-wave contamination during the high-energy wave events, while cov(U, ΔW) is arguably worse than cov(U, W) (Figs. 13b,c). The two filtered estimates agree well (Fig. 13d). These results indicate that the three estimates cov(ΔU, W), cov(ΔÛ, W), and cov(U, ΔŴ) are all successful at removing surface-wave bias when it exists in this dataset and that, as shown by comparison with the raw estimate, the successful techniques do not degrade the shear stress estimates when energetic surface waves are absent.

c. Heat flux

As in the case of shear stress, we consider the cospectra that are integrated to yield the three sets of T′w′ estimates described in section 3, because wave-induced contributions to heat flux are also observable as deviations from the expected form of Co_Tw in typical boundary layer flows. Empirical forms for Co_Tw and the running integral of Co_Tw, ∫^f₀ Co_uw(f′) df′ (Fig. 8b) from Kaimal et al. (1972) are similar to those for shear stress, except that the range of heat-flux-carrying eddies is shifted to higher 2πfz/V in comparison with the range of stress-carrying eddies (Fig. 8a).

For the most part, the bottom boundary layer is well mixed during periods of strong surface wave activity. However, near-bottom stratification existed during Hurricane Hortense (Fig. 5b), which resulted in temperature fluctuations produced by surface waves. The stratification was inferred to result from mixed layer entrainment and was accompanied by physically meaningful buoyancy fluxes [see Shaw et al. (2001) for further discussion]. As an example of the effect of these wave-induced temperature fluctuations on Co_Tw, we present a temperature spectrum and cospectral ogive curves at 3.3 m above the bed, position (1) in this case, for a burst in which the rms wave-induced temperature was 0.01°C (Fig. 14). In this particular example, the surface-wave peak in the temperature spectrum (Fig. 14a) is located at the upper end of the expected range of the heat-flux-carrying eddies (Fig. 8b). The raw estimate contains an anomalous contribution at the dominant surface wave frequency (Fig. 14b). The anomaly is visible in both of the differenced estimates (Fig. 14c) and the filtered estimate cov(ΔT̂, W) (Fig. 14d). Only in the filtered estimate cov(T, ΔŴ) is the observed bias removed (Fig. 14d). The heat-flux-carrying eddies in this example are located in a higher range of 2πfz/V than expected, which is probably caused by stable stratification limiting the size of eddies (compare with Fig. 8b). The pronounced effects of surface waves on estimates of heat flux contradict the theoretical scale analysis presented in section 2a.

Here, the burst containing an internal wave presented in Fig. 12 is considered in terms of the effect of the internal wave on estimates of heat flux. The time series of sound speed from 1.1 m above the bed is nearly constant until disturbed by the passage of the internal wave 16 min into the burst (Fig. 15a). The internal wave made large contributions to the raw estimate, and the contamination is not eliminated in the filtered estimate (Fig. 15b), as for the case of shear stress. In fact, both estimates result in a positive heat flux, which is countergradient for the stably stratified bottom boundary layer. Note that the sound-speed record (Fig. 15a) is inconsistent with the implicit assumption of stationarity of the statistical properties of waves within individual bursts. Unlike Co_uw, Co_Tw is contaminated well into the expected range of heat-flux-carrying eddies, above 2πfz/V = 1 (Fig. 8b), so that Co_Tw cannot simply be windowed to remove internal wave bias without a loss of turbulent flux.